Let’s face it, 90% of everything on the Internet is crap.
So I cannot believe I did not come across these until now. Maybe they were just lost in the swamp of effluvia. I came across so many good resources lately that I am planning on re-designing my course next spring to include a lot more applets, videos and other cool online options.
Here are some sites in the “Dude! You have to check this out!” category
Against all odds – statistics videos created in 1989! These are 26 half-hour videos on different topics on statistics. Think Nova for statistics.
Yummy Math – a website of activities making mathematics relevant to the real world. That’s their tag line. I’d say they make mathematics INTERESTING, like figuring out your savings buying Christmas presents, or comparing the durability of twinkies and tomatoes through time lapse photography or computing how much coffee could fit in a giant coffee cup. Go there. See for yourself.
SAS Curriculum Pathways – an enormous free site that has an unbelievable amount of stuff on statistics, algebra, geometry – oh, yeah, and I guess English, Spanish and Social Studies, too (if you care about that stuff). I have no excuse not to have looked at this before because I have been hearing about it for years and the nice people from SAS sent me links which I never clicked on but just sent to friends of mine teaching middle school and high school. Hey, I’m busy. That’s no excuse. The school where I volunteer has a shortage of textbooks. Well, this site has pages to read, then research questions, then statistical applets.
Not strictly set up as an educational site, but Policy by the Numbers blog is like my twitter stream but far more in-depth. Posts on open data, Google hang-out on AP statistics. It was educational for me. Made me want to teach high school AP statistics. Just listening to this one video gave me two new books I want to read as possible textbooks for my class, so it’s educational for me.
The mathalicious blog is really cool. Their site also seems to have some good activities based on the sample ones but I’m not sure because they want $185 a year for a license, which strikes me as a bit steep.
Last but definitely not least is CAUSEweb.org – which I had actually seen before but I guess I was busy (detecting a pattern here?) This is the Consortium for the Advancement of Undergraduate Statistics Education. The first thing I saw here was a free workshop in San Diego on playing games to teach statistics, funded by a National Science Foundation grant. I signed up for it on the spot.
It’s been a very productive week for finding sites and other resources I want to review for teaching statistics. So much so that I used this other site, 43things.com to make a list of all of the stuff that I want to consider for the grand-a-mundo course re-design.
Please, please, please if you have suggestions, chime in.
Here is a math problem:
Hoksinato and Tasunka Ska are going to steal horses. They could steal the scrub ponies from the edge of the camp. The last 15 times warriors from the tribe tried to steal scrub ponies, they got away 12 times and were caught and tortured 3 times. If they steal the war ponies instead, they will show more bravery, maybe even earn an eagle feather. Plus, war ponies are much more valuable. The last 10 times warriors from the tribe tried to steal war ponies they got away 3 times and were caught and killed 7 times. What is the probability that they will steal the war ponies and get away?
The correct answer is 30%, in other words, 3 out of 10 times.
Another way you could possibly answer it is 12%. You could interpret it as there were 25 attempts at stealing ponies, and that the question was,
What is the probability that they will steal the war ponies AND get away?
In that case, the probability is 40% that they will steal the war ponies (versus the scrub ponies) and 30% that they will get away, .40 *.30 = .12
Thinking about this, I decide to re-word the question. To say,
Hoksinato is not sure whether or not they should try to steal the war ponies. What is the probability that warriors stealing war ponies will get away?
A statistician would immediately think of this in terms of P(A|B) in other words, what is the probability of escape given that they are stealing war ponies? The answer is clearly 30%.
Here is why solving this problem is hard
- It is not worded to be completely clear whether I need to know the probability of getting away when stealing war ponies or the probability of getting away AND stealing war ponies.
- I need to decide which numbers are relevant. For this particular problem, the probability of escaping when stealing scrub ponies is irrelevant.
- I need to decide on the correct operation, in this case, to find the percentage of successful war pony theft attempts, which I find by dividing ten into three.
- Finally, I need to know the answer to 3/10
Why don’t I just ask, “What is the probability of escaping, if a warrior escapes 3 times out of 10?” Because that is a much easier problem. Here’s the kick in the ass – problems in real life don’t come up that way. They occur ambiguously worded with extraneous information thrown in.
Many people, including most of those awarding funding at the National Science Foundation, realize this and thus very strongly urge mathematics programs to teach problem-solving via discovery learning, guided discovery or other methods. These people are right – to an extent. As you can see above, basic mathematics alone won’t solve this problem.
Many other people, including many math teachers at low-performing schools, believe the NSF is run by COMPLETE IDIOTS because the students don’t know the concept of probability, much less P(A|B) , they have no idea of the notation, where to even begin deciding which are the relevant numbers and oh, yes, they can’t divide 10 into 3 and come up with .30 either. These people are also right – to an extent.
If we are going to teach kids math effectively, we need to fund projects that bring these two sides together.
Y’all get on it.
It’s not often that you read a paragraph and it sticks in your mind for months. That this particular paragraph came not from some great literary work but rather from the proceedings of the annual meeting of the Association of Small Computer Users in Education is even more expected, but there it is. Douglas Kranch wrote:
“Expertise develops in three stages. In the first stage, novices focus on the superficial and knowledge is poorly organized. During the end of the second stage, students mimic the instructor’s mastery of the domain. In the final stage, true experts make the domain their own by reworking their knowledge to meet the personal demands that the domain makes of them.”
Existing mathematics (and statistics and science) education programs are too limited. They either focus ONLY on drill and practice, not progressing past the first stage, or they try to skip the first stage or two entirely, an overreaction that while having the laudable goal of teaching “higher order thinking skills” often leaves students frustrated and discouraged as they do not have the basis for the tasks required. Part of the problem comes from, I think, having subjects taught by people who were not experts themselves.
Let me give two examples, one horrid and one good.
It’s common for middle school teachers to give students assignments that are supposed to be “relevant”, for example, “Make up your own periodic table”. They did not, however, come up with a new way of arranging elements. No, they did a periodic table of football players or TV shows. I suggested to The Spoiled One that perhaps she could do Disney channel shows and have those that had a character move from one show to another be in one group, just like elements that lose an electron are in one group. Similarly, those shows that shared a character, like if Miley Cyrus also did appearances on The Suite Life could be a different group, like those elements that shared an electron. I was out of town at the time – (if you follow this blog, you know that my children contend most stories of their childhood begin this way) – but when the project was due, the world’s most spoiled thirteen-year-old turned in something she drew with a pencil on a piece of paper and got a 50% on it. When I quizzed the rocket scientist about how this happened he answered unrepentantly,
“I didn’t make her put any effort into it because she said it was stupid and I agreed.”
While I did not have this precise conversation with the school – Seriously… What. The. Fuck – you want a kid to learn about the periodic table, covalent and ionic bonding – you teach them that, NOT relate it to stupid TV shows we’d just as soon she not watch any way. We spent many hours going over with her the idea of electron shells, what happens when a shell is not full, the number of electrons in each shell. You want a kid to know that NaCl is sodium chloride? You explain that Na is the symbol for sodium and Cl is the symbol for chlorine and you put the two together and you get sodium chloride. There’s actually some really interesting stuff you can throw in there about how it’s kind of weird that when you combine these two elements you get something that really isn’t very similar to either one individually. You want to get kids interested in chemistry? Do experiments. Few things are more motivating to the average eighth-grader than the possibility (however slim) that they might get to see the school blow up with the teachers in it.
I’ve been a teacher. I started out, like most people, as a not particularly good teacher, and then, with years of experience, I got better. I recognized that all of that stuff, like the periodic table the electron shells, multiplication tables, how to read an ANOVA table, you need to learn that. Even if you don’t get it at first, if you ” … focus on the superficial and your knowledge is poorly organized” – you still learn that p-values, df, sums of squares should be in there. At first, you don’t know what df stands for and when you find it is degrees of freedom that doesn’t really tell you much. After a while, you vaguely start to get it. It’s frustrating, it really is, going through those motions you don’t really understand – but there isn’t any alternative.
Yes, in this process, I drew a lot of connections to other programs I had written in other languages. What I did not do is draw a parallel with the time we got lost and went driving around Miami trying to find somewhere it was legal to make a u-turn. (Let me just say that Florida has commitment issues. If you are going south they think you should just keep going and if you change your mind and want to go north, forget it.)
What I also did not do is one mindless fill-in-the blank or multiple-choice exercise after another ad infinitum. I didn’t memorize rules until I could pass some arbitrary test at 100% accuracy. Although I did start with that, I didn’t finish with it. In fact, I did the very minimal amount until I could move on to the imitating experts and making it my own.
If you want to learn programming, statistics, chemistry then DO that. Don’t just read about how to do it and for the love of God, don’t do something else, like stupid charts of TV shows or biographies of women mathematicians and pretend you’re doing STEM education.
My mother was the first American to win the world championships, so I called her for advice, and believe me, Mom is always brimming with advice, whether you want it or not …
In fact, all parents have the experience that their own children occasionally take advice from strangers far better than from them. So, for your daughters or mine, here are three pieces of advice on succeeding in the tech world.
1. Learn Calculus – Ignore every person who tells you that you won’t need it, it’s too hard. Take it in high school and take it again in college. People often say, “I just can’t do math.” That’s bull shit. You just can’t make the NBA. You can certainly do math. My youngest daughter whines that way sometimes and yet she doesn’t sit and read her Algebra book unless we stand over her and make her do it. Here is why you need to learn calculus:
- 99% of all math books are written to be so boring that you want to track down the authors and bitch slap them. Learning calculus is good training for life because there WILL be boring things you have to do to get to where you want to be. By ten years old, you should have overcome the idea that everything has to be as intrinsically rewarding as laying on the couch with your puppy.
- If you do happen to have a hard time, even better. Don’t skip class. Read the textbook. Get a tutor. Read the book again. Everyone at some point runs into concepts that are difficult. This happened to me twice, in calculus and in my fifth semester of statistics in my doctoral program. Now, in both areas, it is hard for me to understand why it was ever confusing, but I remember at the time reading the book over two or three times and still being a little fuzzy and afraid I wouldn’t ever get it. I’m married to a real-life rocket scientist, a man who decided to pursue a Ph.D. in particle physics because “nuclear physics was too easy” and even he had a point in school when he ran into concepts he had to read over and over until he understood it. Find a way through. That’s a super-important lesson in life.
- If you learn calculus, you WILL use it. I took Calculus I & II my freshman year of college. It came up in a few economics courses my senior year and in my first statistics course, which I took in the math department. I never had any use for calculus for years after I graduated. Then I went on for a Ph.D. and specialized in Applied Statistics. Calculus was really useful in some of those courses. Now, in my profession, whether reading research, reading documentation or programming, it comes up fairly often. Not every day, but certainly every month.
2. Learn to say “Fuck you” and say it both openly (rarely) and to yourself (often).
My friend has a reputation for a great bedside manner. He uses a code phrase. When a patient says something like:
“I have decided to treat my cancer with grapefruit juice instead of chemotherapy.”
“I understand how you can see it that way.”
This is his code for,
“You’re a fucking moron.”
You need a code phrase because people will try to dissuade you, denigrate you and generally provide useless advice (contrary to the wonderful advice I am giving you now). They will tell you that you cannot be an entrepreneur because you want to have a family. They’ll tell you that you are not a real ‘techie’ because you don’t have a degree in engineering. If you do have a degree in engineering it will be because you don’t have a degree in Computer Science. If you do have both degrees and have experience as an engineer and programmer it will be because you don’t know a specific programming language. Some people seem to have a sadistic desire to pull other people down, saying things like,
“You may have a masters degree but it’s not computer science from MIT. You don’t program in Ruby or Java and everyone knows that unless you have years of experience in both of those you are not really marketable.”
Feel free to tell those people, either:
“I understand how you can see it that way, but I’m going to go ahead and apply for the position at the accelerator anyway.”
“Fuck you! I’m going to do it anyway and I don’t care what you think.”
Seriously, there are very few insurmountable obstacles. One of my daughters received a Fulbright scholarship to study in Germany for a few weeks. She almost didn’t apply because she had a young child. I told her that she was being ridiculous, she had a husband, a mother, a mother-in-law and two adult sisters. Between the lot of us, we could take care of one baby.
So what if I let her teethe with Twizzlers during the week I was there?
I also took her swimming in the hotel pool every day, to the science museum, to the aquarium and taught her to dance in elevators. And when Maria came back from Germany, her daughter was still alive, better than ever, because, hey, she had a couple more teeth.
This is really the most important piece of advice I have. Don’t let anyone discourage you and that includes yourself.
3. Learn a programming language or two.
If you followed my first two pieces of advice, this third one will be easier. The whole trick to learning a language is to not get discouraged and plug away at it. Read a book. Write some code. Read another book. Look at programs other people wrote. Think of some things you want to do with that language. Try them. Fail. Swear. Try again. Don’t get discouraged.
” Expertise develops in three stages. In the first stage, novices focus on the superficial and knowledge is poorly organized. During the end of the second stage, students mimic the instructor’s mastery of the domain. In the final stage, true experts make the domain their own by reworking their knowledge to meet the personal demands that the domain makes of them. “
This is why those first two bits of advice matter. In learning programming it is easy to get bored or discouraged as you go through those first two stages. It’s easy to start believing it’s too hard, that guy who told you women don’t have the same natural talent for programming was right, it’s too late for you to start now because you didn’t take enough math in college …
“I understand how you can see it that way.”
I was very busy this weekend working on the semi-annual site update (I am SO getting last place in the Search Engine Optimization contest) and starting on my book – Beyond SAS Basics: Tips, Statistics and a Naked Mole Rat and on TOP of all of that, I had to take the world’s most spoiled 13-year-old shopping because there are apparently some items of clothing and footwear existing in Santa Monica that she does not own yet.
I’m also working on a proposal for math education software and I got to thinking that there is SO much out there, how can there possibly be the need for any more. In (very) partial payment for the shopping spree, I had The Spoiled One review math games and websites for me. Since I don’t see the need to call out any particular resource just because she happened to randomly land on that one today, the names have been omitted to protect the guilty.
As background, I should tell you that she was recently accepted for a summer program for high-achieving girls, scores above average on standardized tests for math (not as above as WE would like) and has never made a grade below a B in anything. (Because in our house a C means you are grounded until the next report card.) On the other hand, homework is sometimes accomplished only as a means to effect the return of all of her confiscated electronics. In other words, she is a little better on achievement and motivation than the average student, but hardly a paragon of mathematics virtue. And here were her reviews:
Video of Rap Song on Mathematics Topics (Because, you know, you kids these days like that)
… Um, distracting. I learned nothing because I couldn’t understand the lyrics.
Place Value Video Lecture
Not really for someone my age (13). Kind of stupid anyway.
Sucks! (She drew a picture here to indicate how much she hated it.) BORING. Doesn’t really work. (Punctuated by another picture)
Game with Word Problems
The game was good I guess … (a few minutes later…. ) Never mind. It didn’t give you the right answer after. I HATE THIS SITE.
Game on Factors and Multiples
OK. Not creative or fun. (Another picture, that looked something like this
- . .
Sites on Math in Every Day Life/ Real Life Math
Eewww NO!! Doesn’t make me like math!
Mucho Math- The only one that didn’t suck
“That one with the Hispanic math teacher and the kid. That one was okay and kind of funny even though the topic it was on wasn’t really at my level.”
I found this last comment extremely interesting because I knew who she meant. I had sat my daughter down at a computer on a web page with over 1,000 videos, games and other math resources and she came up with the same option that I thought was one of the best ones I’d reviewed when I was doing the same thing a couple of months ago. The teacher is Lawrence Perez. The innovation he has included is really quite simple – he has a student in his video.
Having reviewed numerous other options myself, I have to say I agree with my daughter on much of it. The absolute WORST thing you can do in designing mathematics software is have it get the wrong answer, for example, when it asks :
If Y = 5 + x**2 and Y = 14 what is X
and you put -3 and it says
WRONG! The answer is 3
Of course, -3 is also a valid answer and then you have a student who says,
“I hate this program. It sucks!”
Not as bad, but also frustrating are those programs that don’t tell you the answer, but simply come up with the next question.
If you say that both of these problems are examples of poor design, well, I agree with you, but poor design seems to be rampant.
Having a game or video that is too basic is not the problem of the software, of course, but MAYBE whoever marketed it as being at the middle school level. Or, it may just be that there is wide variation among students and was not appropriate for this particular student.
Yes, I’m generalizing from an N of 1 (well, 3 actually, if you include me and my brother, who is a math teacher and has had generally the same responses), but from what I have seen so far, there is a whole lot of math education software out there that is not effective in interesting students enough to use it. Sometimes the game doesn’t even do the minimal job of providing the right answer, something any parent could accomplish with a $1.29 stack of index cards by writing the question on one side and the answer on the other.
Every time I have done this experiment, whether with me, my daughter or someone else, the outcome has been equally underwhelming. Even more underwhelming is the fact that almost NONE of the designers/ producers of these resources even MENTION the thought that perhaps one would evaluate the software and see if it has any impact at all. The attitude seems to be “Here you go”. Period. Kind of depressing.
I guess the good news is that there are about a bazillion more games, videos and other resources out there to try.
A lot of sources kind of sucked. At best, these sites were just the same old thing, flash cards but on a computer screen, for example. There is nothing terribly wrong with that, but it is hard to imagine that they have any greater benefit than just using index cards you picked up at any store and writing 2 x 3 on one side and 6 on the other, which is how I think everyone has learned multiplication since we quit writing on slates with a piece of chalk. Hey, maybe we should go back to that. It probably involves less waste. Green math! But I digress …
At worst, these sites were just plain wrong. This was more often true for those that dealt with less basic mathematics, where they would, for example, give a definition for a chi-square that was really for a t-test or say that the median was the most common score in a distribution (It isn’t. That’s the mode.)
Other sites were better, including videos of short lectures and the explanation of whatever the topic they were teaching was correct. (AnnMaria’s first rule of teaching – have something non-stupid to say).
Two examples are:
- Khan Academy site, which is free, has over 2,000 videos and Bill Gates as its BFF.
- Cool Math Guy website has some free samples, for others, you have to pay. The videos I saw are good explanations of such topics on trigonometry.
There are hoards of math game sites out there, many of which are just a computerized version of asking your child over and over what is 47 + 52 until his brain crawls out his left ear and runs away just to escape the boredom.
Then, there are sites like Gamequarium, which offers a LOT of different math games for every topic, most of which look like they would be fun if you were immature, which I am.
ALL of the resources I found suffer from the same fatal flaw which is that they begin with the presumption that the student has some interest in learning math. This seems a reasonable, some might even say ‘sane’, assumption based on the fact that the person has come to a site that is for teaching mathematics. For those people who seek out these sites, they might work.
The problem is with the vast majority of people who WON’T ever voluntarily go to these sites because they really don’t give a rat’s ass if they ever learn math or not. Sometimes, as this excellent article “The Education of Jose Pedrazza” points out, they are much more concerned about whether they are going to be homeless, how their family is going to eat.
Given those circumstances, it’s really hard to focus on if you learn this math, you’ll be able to do next year’s math and so on for the next 10 years until you graduate from college and get a good-paying job. It’s all well and good to talk about delayed gratification when you are sitting here like me drinking Chardonnay at an expensive oak desk, and quite another when your mom is collecting cans to come up with money for dinner.
Some of it, the odds are great that you will NEVER use. I just came across this statement in a publication on research in teaching and learning mathematics.
“Across all age levels, the best estimates are made in temperature situations and the most difficult estimates involve acreage situations.”
ACREAGE? Okay, I’m 52 years old, I use math for a living, I’ve bought and sold four houses in my life, including one that had five acres of land with it and was in North Dakota AND NEVER IN MY LIFE HAVE I NEEDED TO ESTIMATE ACREAGE!!
Yes, I am sure there are farmers and landscape architects and people doing surveillance for homeland security applications who may need to estimate acreage. Every time I write something like this, I get hate mail from people telling me this is why they will never hire me to work for them at Google Maps. (Of course, when I look up these people, they never actually work for Google, or anybody. They are invariably some embittered graduate student teaching Mathematics of Acreage Estimation at Boo-hoo U. )
My point is that most of math is taught completely out of context with no real thought to application other than answering a question on the SAT. For some students, like the most spoiled 13-year-old in America, who happens to live in my house, that is adequate enough incentive. One reason is that for her, and many of her peers, it is NOT gratification delayed ten years. At the end of the school year, many neighborhood parents trek to the Apple Store to buy the iPhone 4 or the gadget du jour for Buffy and Justin who got an A in math. In eighth grade, the kids will all take their high school entrance exams, and when the test scores come and acceptance letters come out, there will be ANOTHER round of iPhone -buying and trips to The Grove. A couple of years after that, many of those same kids will get their first car, with the stern admonition that, “Your grades better stay up or you will be walking to St. Alphonso’s Catholic High School “.
I was a little depressed after I read this article on the Los Gatos Patch, where the mother happily admits that she could not do her 14-year-old son’s Algebra class. It tells me not only that we find it perfectly acceptable not to know math (while it is NOT okay to say that you forgot how to read) but also that the mom obviously has no need for Algebra in her daily life. On the other hand, I was majorly impressed that she got her son to make dinner and to clean up – twice.
Some people just like math – I did and I still do. That’s only incentive, though, to study the parts that interest you. For example, I watched a video on trigonometry for about five minutes. Then I was bored. It was exactly like the movie, Freaky Friday, where the middle-aged mother changes bodies with her teenage daughter, and in algebra class tells the teacher, “No, believe me, I will NEVER use this.”
I use algebra nearly every day of my life. I use matrix algebra, not every day, but certainly weekly, and calculus fairly often, too. On the other hand, I have NEVER and I do mean, NEVER, needed to know a sine, cosine, tangent, arctangent for any reason whatsoever, not even when I was an industrial engineer. This isn’t to say that no one ever uses these. I asked the house rocket scientist when was the last time he used any of these and he said that everyone in the real world uses all of these every day. Well, EXCUSE ME!
Perhaps we have it backwards. Instead of railing about the poor performance of our kids on tests and teaching to the test, maybe we should turn things around. Perhaps we should start with why they need to know how to calculate acreage, t-tests or cosines. Give them some projects where this information as applied. Maybe then not only will they actually give a rat’s ass if they learn it or not, but they’ll also still remember it when they have 14-year-old kids of their own and be able to use that information on the job when people like me hire them.
Wouldn’t that be a nice change of pace?
“If everyone knows a thing it’s almost for sure it aint so.”
“It’s not so much the things you don’t know that hurt you as the things you know for sure that aint so.”
I don’t take anyone’s word for anything. Take those quotes, for example, which I’ve both heard attributed to Mark Twain, Will Rogers and several others, ironically by people who were just certain they were correct.
One thing everyone knows is that Americans suck in math. We are so far behind Asia, we are continually told, that we are soon all going to be learning how to say, “Would you like fries with that?” in Chinese.
There was an article in the Los Angeles Times today that profiled a mother who had an Excel spreadsheet with a schedule for her child from 8:00 a.m. to 11 p.m. seven days a week. She said she started in kindergarten, because life is hard and students need to learn to deal with it. Her son, as a tenth grader, scored a perfect 800 on the SAT. Rather than convincing me further that we suck at math, it made me question the goal of propelling a child to perfect scores.
One thing writing a dissertation on intelligence testing taught me is that test scores are very, very far from absolute and objective. Two critical points to keep in mind:
1. Some group of people decide what is tested, inevitably the group of people that has the most power. If we insisted that being fluent in more than one language is a factor in achievement scores, Hispanic children would be getting admitted to elite institutions in droves. Before you discard this as a silly notion, think about the arguments made for including high math scores – these are relevant to courses students take, to careers. An argument could be made for functioning in a global market place, for the ability to read texts in the original Spanish (or whatever second language a student reads). I could write a whole dissertation on this – oh wait, I did ! – the point is we make decisions about what goes into the tests and those decisions favor some people and not others.
2. The scores we use to evaluate both at an individual and larger (school, country) level are almost never how many questions were answered correctly, which you might logically think is your test score. There you go with the logic again. Cut it out. In fact, scores depart several steps from the number of correct answers. First, there is the issue of partial credit, yes or no and if yes, for what. Second, there is the step of standardizing scores. Usually this means setting the average at some arbitrary value, say 100. If the average student gets 17 questions right, then that is set as a score of 100. The standard deviation, the average amount by which people differ from the average, is also set at an arbitrary value, say 10. (If you’re not familiar with these ideas, think of your family. We’re kind of short in my family, and if you went to a family re-union you’d probably find that the average woman is around 5’3″ give or take two inches. So, you can think of five feet three inches as being the average and two inches as being the standard deviation. If you are reading this and from a country on the metric system, 5’3″ is equal to a furlong plus a bushel, a peck and a hug around the neck.) To return to my long-forgotten point – if 84% of the people score 22 points or lower, than answering 22 questions correctly is given a score of 110. (The mean of 100 + one standard deviation of 10). The scores you see reported aren’t that closely related to the number of questions answered correctly and they tell you almost NOTHING about what precisely people do or do not know.
I think most statisticians know this. I am certain that nearly everyone who does analyses of educational tests knows this. But I am equally certain that the average person reading the newspaper does not. This is important because it has to do with our sucking or not.
My assumption, based on what I read in the papers and hear on TV is that American kids just don’t know basic math. So, I downloaded the TIMSS (Trends in International Mathematics and Science Study ) data and I also downloaded the items that had been released, to see what it is that American kids do and do not know. Here are a few examples:
Students were shown a rectangle divided into twelve squares. Five of those twelve squares were shaded. Then, they were given five choices of circles that were partly shaded and asked:
“Which circle has approximately the same area shaded as the rectangle above?”
To solve this problem you need to figure that the rectangle has 5/12 shaded and understand that 5/12 is a little less than one-half. (The figures show a circle that is 7/8 shaded, 3/4, exactly one-half, a little more than one-half and a little less than one-half.)
This question was answered correctly by 80.2% of American eighth-graders.
The next question asked :
A gardener mixes 4.45 kilograms of rye grass with 2.735 kilograms of clover seed to make a mix for sowing a lawn area. How many kilograms of the lawn mix does he now have?
This question was answered correctly by 71.9% of American eighth-graders.
I must admit that I was surprised the figure was that low, although not extremely surprised, since I know many, many adults and some young kids who never do math like this. Every phone, every computer has a calculator on it and they just think this is a useless skill, like cursive. I happen to disagree and the world’s most spoiled thirteen-year-old is not allowed to use a calculator to do or check her math homework.
Another question dealt with inequalities:
X/3 > 8 is equivalent to….
To get this answer correct, you need to understand the idea of inequality and how to solve an equation with one unknown. Essentially, you need to reason something like 24/3 = 8 so X > 24 . This, of course, presupposes you also know that 24/3 = 8.
This question was answered correctly by 42.8% of American eighth-graders.
A question that was answered by even fewer was:
What is the perimeter of a square whose area is 100 meters?
To answer this you need to know:
- The formula for finding the area of a square
- The concept of a square root
- That the square root of 100 is 10
- The area for finding the perimeter of a square (or rectangle, either would work).
This question was answered correctly by 26.5% of American eighth-graders.
One last question,
A bowl contains 36 colored beads all of the same size, some blue, some green, some red and the rest yellow. A bead is drawn from the bowl without looking. The probability that it is blue is 4/9. How many blue beads are in the bowl?
This question was answered correctly by 49.4% of American eighth-graders.
Are these percentages bad or good? Honestly, I thought the questions were pretty easy and I was surprised by the low percentages on some of them – but I do math for a living and I was in 8th grade almost forty years ago. So, I have known this stuff a very, very long time. I THINK some of the questions were actually what was taught in ninth or tenth grade when I was riding a brontosaurus to school, so the fact that eighth graders today don’t know this information doesn’t convince me we’re all a bunch of drooling idiots.
Here is a blasphemous question for you - Does it matter if you know the answers in eighth grade? I’m serious. Is it worth having your child study from 8 a.m. to 11 p.m. so that he or she knows all of this in the eighth grade instead of the ninth grade?
A few weeks ago, I was looking for data for a proposal I was writing and came across a state Department of Education website that had a note on its pages on test scores that said proficiency meant something different according to the federal government definition and that many people could function perfectly fine will being scored below proficient in math.
At the time I dismissed this as an excuse for poor performance. Today, when I looked at the questions and the results, I was not so sure. My two older daughters are a journalist and a history teacher. Both have degrees from good institutions (NYU and USC). I believe neither of them could answer the question about finding the perimeter of a square with an area of 100. Perhaps they could have answered it when they took their SATs or while they were taking the one mathematics course they took as undergraduates. I’m not sure. I’m fairly certain if they ever knew this information, they’ve totally forgotten it. The truth is, as much as I hate to admit it, that neither of them at any point in their lives will feel the lack of this knowledge.
On the other hand, my daughter who knocks people down for a living (she competes professionally in mixed martial arts) could almost certainly answer these questions off the top of her head, just because she likes math and has always been good at it.
What percentage of Americans (eighth-graders or not) SHOULD be able to answer these questions?
I have no idea what the answer to that is.
Some people would say 100%, because they need to know this information to do well on the tests to get into a good college. I’m not sure that is true. More and more, people are asking WHY you need to do well on the tests. If I want to be a sportswriter or a history teacher or a doctor, what good does it do me to be able to calculate the perimeter of a square given the area?
I think the mother in San Marino may be part of an education bubble that will burst just like the housing bubble has. I am far from the only person to be suggesting this. Not only has the cost of higher education reached astronomical levels where it exceeds the cost of a home in most parts of the country, but it also, for selective institutions, is costing more of your life. Not only are fewer people going to be able to pay it, but, perhaps like the housing bubble, more people are going to say, “This isn’t worth it.”
I did not work from 8 a.m. to 11 p.m. I spent several hours today reviewing grants. Then I went running down to the beach, because it was a beautiful day. I had a Corona while reading the LA Times. I analyzed the TIMSS data and I watched The Daily Show. I also checked my daughter’s math homework and pointed out the one answer she had incorrect. She figured it out and fixed it on her own.
Life is not hard. Life is good.
A couple of nights ago, I had a nightmare. I dreamed that I couldn’t do math. I was having lunch with some colleagues and the bill was $24.82. Everyone handed me money and I had $25.67. I was trying to subtract the bill amount from what was in my hand and divide it by three, but I couldn’t. Every time I thought I started to have the answer, the numbers flew right out of my head. Since it was a dream, I could see them flying, with little wings and everything. As time passed, my colleagues started to get impatient, ask me if I was done yet, make jokes. I remembered that book, Charlie, and started thinking, this is what it must be like to be mentally retarded. I was so upset, I woke up.
I’ve been slacking on the reverb10 project. I read about it and it sounded interesting. The idea is that every day there is a different prompt and you’re supposed to post on your blog related to that. I have a blog. Three, actually, though that’s another, unrelated story. I thought it would be good for me to write more, since, oddly, I often learn things better as I write about them. Well, it has been really interesting, but in a different way than I thought.
As I read the prompts, and the other bloggers responses to them, I was very strongly reminded of Sheila Tobias’ book, They’re not dumb, they’re different: Stalking the second tier. In brief, her book is about her study of why very bright people nonetheless choose not to study science and why they have a hard time with it. She had scientists sit in on literature classes and people with doctoral education in subjects like English sit in on introductory science classes. It was a really fascinating study and reading it, I could totally identify with the science Ph.D.’s frustration with English 102. It was just like Dave Barry said about college, that he chose English as a major because it had no actual facts in it, unlike Chemistry, where they get really snippy if your chemical formula for, say, what happens when you combine two hydrogen atoms and one oxygen comes out to be really different than everyone else. If you say, “Maple syrup!” or “The Queen of England”, they do not give you points for creativity, quite the opposite.
I tried to avoid every single art and humanities course in college. I did take Japanese as a language, since I went to Japan to study for a year. Since mathematics was in the College of Arts and Sciences, that took care of that distribution requirement. They caught me my last semester in my senior year and made me take English Comp, which I managed to do as an independent study with a sympathetic English professor.
So, I looked at the reverb10 prompts and did not do that many of them. I wasn’t quite sure they were talking to me. For example, when the prompt was about what you appreciate, it occurred to me that I appreciate Euclid, logistic regression and my husband, not necessarily in that order. My suspicion that I was playing on a team by myself here occurred when I typed reverb10 and logistic regression into Google and all the hits that came up were me.
So, I’ve been reading these posts by other bloggers and I truly feel like Temple Grandin in Oliver Sacks book, An Anthropologist on Mars. I read this blog by a 20-something person who feels guilty about not meeting with people she used to know. The same blog had a link to an awesome article on a man who decorated his basement with $10 worth of Sharpies. Awesome for him, but I’m guaranteeing you that if I tried that my house would just look like Matt Groening or Hugh MacLeod went completely psychotic.
It reminded me of The Perfect Jennifer when she was about nine years old deciding she wanted to teach herself to play The Sting, by Scott Joplin. So, she got a copy of the movie with Robert Redford and Paul Newman and played that part of it over and over until she could play the song by ear. I couldn’t imagine ever even thinking of wanting to do that, much less doing it. Even though her dad had died recently and I did not have a lot of money, I went out and bought her a piano.
There was another reverb10 prompt on what have you made this year. I thought to myself, “Does dinner count?”
Lots of people had made lots of things. some of them, like basement-Sharpie-guy, just amazing, and others that you could have bought at the dollar store made by some kid in China and I didn’t want them anyway.
So, I typed in “math” and “reverb10″ and came across an interesting blog by a math teacher who quit her doctoral program to go back to teaching. Even though I did finish my doctorate, and, in fact, enjoyed it, I could totally related. Jane Mercer, one of the people on my doctoral committee, and a profound influence on my life, had a sign in her office that simply said,
“No matter how far you’ve gone down the wrong road, turn back.”
Then, her next post was about making wreaths out of buttons and I thought,
“Why would you even do that? No, seriously, why?”
And it occurred to me, because I am not really all that slow on the uptake, despite my nightmares, that there are some people who would think the same about me.
Tomorrow, when I am sitting in the airport, I am going to write a blog about quasi-separation and other problems with logistic models. I’m really looking forward to it. Usually when you read papers on some statistical procedure they have these stupid, perfect little datasets that are set up not to offend anybody so they are something like the auto.dta dataset from Stata, and everything works out perfectly to be highly correlated with no problems of multi-collinearity and the chi-square is always significant and the R-square is always really awesome and something like .80. So, you get graduate students who have an R-square of .42 for their dissertation data and they are disappointed instead of simultaneously having orgasms and doing the little happy dance like the situation warrants.
My paper is going to start out early on with real life, like getting a chi-square with the probability > .97 and the “NOTE: This model may not be valid” on your output, which causes you to comment to yourself,
“Yeah, no shit.”
In writing this paper, though, I am really, really trying to keep in mind button-wreath-woman and basement-Sharpie-writing-guy and person-who-feels-guilty-over-coffee and think what would make it interesting and relevant to them. I think I will write better papers in the end.
So, that’s what I learned from reverb10.
My friend, Gokor Chivichyan, a mixed martial arts instructor, once gave this questionable advice to his students,
“Women usually just say they’re fat to get attention. So, me, I agree with them. If she says she’s fat, I say, yes, you fat but we like you anyway. If she’s really fat, though, you just have to dump her. Not if she’s your wife, though. Then, it’s too bad but you have to keep her anyway and take care of her because she’s the mother of your kids.”
(For the record, I have met Gokor’s wife and she is both lovely and charming.)
We tend to keep our private life extremely private. Dennis has been referred to as “your alleged husband” by my friends, who have never met him. Recently, another friend of mine, my former business partner from Spirit Lake Consulting was visiting and, after knowing me for 20 years, met my husband for the first time. My friend commented,
“Your husband really loves you.”
I was a bit miffed that he found this so surprising. I think one reason this is a surprise is that we so often categorize people into single boxes. I was a world-class athlete while my husband hates all exercise. Being a sixth-degree black belt, I once suggested to him that perhaps he could learn judo for exercise. His response was,
I’m not doing anything where people touch me unless I get to have sex with them at the end of it.
As I was laying in bed this morning with my eyes closed, trying to avoid morning, Dennis was carrying on about the Complete Works of Euclid, which proofs were not really proofs, but axioms, the incompatibility of irrational numbers with early Greek geometry, the inefficiency of using geometry for certain proofs rather than algebra or calculus. This is why my husband loves me. Not only did I not find this boring and throw a pillow at him, which most of the women of my acquaintance might have done, but actually opened my eyes, made a comment or two about how it fit exactly with what I was thinking about, which happened to be …
The geometric concept of a line is that it extends infinitely in both directions. What most people think of as a line, with two end points, is really a line segment. Most of us learned this in high school or middle school and don’t really think about it much. However, sometimes it becomes relevant.
Let’s say you were trying to predict a dichotomous dependent variable. Since it is around Christmas time, let’s pick whether a person is traveling home for the holidays or not, which we have coded 1= no, 2 = yes. That might be a very useful fact for people to know who were in either the travel or family therapy industries to target their marketing/ determine outpatient clinic hours.
This is a dichotomous variable and you can see that it plots pretty terribly against a continuous predictor variable – say, income.
Y = a + bX
is just plain wrong here. It doesn’t fit. Very, very far from the assumption that a line extends infinitely we are stuck with a stupid line that goes from 1 to 2.
How about probability then? We could use the probability of going home for Christmas by income. That will extend from 0 to 100, which is certainly closer to infinite.
Well, this is better. It sort of approximates a line.In the graph above, you can see the obtained regression equation
Y = -.1236 + .0313X
(I know you were dying to know.)
You can also see the predicted values it gave me for incomes below $5,000 are negative. I guess those are the people who are not coming home even IF hell freezes over. The probabilities for people with incomes over $40,000 are above 1.0. I guess that means they are going home twice, once to mom’s house and once to dad’s place in the Hamptons with his trophy wife.
So, we have one case, with just the binary outcome, which is clearly not linear. We have another case, predicting the probability of the outcome, which may be linear, but is actually a line segment and not a line. That may be true in theory for lots of things. I doubt income extends from negative infinity to positive infinity, although Bill Gates and Warren Buffet are doing their bit to extend the right side of the distribution for themselves while the Republicans and certain banking and investment firms are making a best effort to extend it on the left for all the rest of us.
There are a whole bunch of reasons that using linear regression is wrong when you have a binary dependent variable, and the fact that it is flat not a linear relationship is just one of those.
Now, if I were an ancient Greek, I would include a lot of geometric examples, not really proofs. If I were an ancient Greek that had access to JMP 8 software I might include another variable graphed against probability and say, “Looky here”, or however you say that in Greek.
This is a very important point – Greek or not – even though the relationship charted above is very high – R-squared = .78 to be precise, it is clearly not a linear relationship. It is an S-curve and it looks very much like a logistic relationship.
Three very important points emerged from this:
- The potential to teach kids the basic understanding of some of the more abstract concepts of mathematics by pictures. I can see how you could start with these graphs and do a linear relationship, then log one variable, log both and start to see the different types of pictures. Those Greeks were on to something. Too bad they didn’t have JMP. Never know what they could have achieved. (Click here for link to random JMP page.)
- The idea of using graphs to teach students is intriguing, and yet I am puzzling how I could drag the world’s most spoiled twelve-year-old away from the Disney channel downstairs and get her to see it that way. The use of graphing calculators in mathematics is not new, but neither does it seem to be particularly effective. This is all fascinating TO ME because I see the end point of making predictions. Perhaps we should spend the first few weeks of mathematics on why what we are about to do is important?
- I was thinking that I had failed miserably on most of the #reverb10 prompts because, well, frankly, I’m more interested in examining logistic and linear relationships than ruminating on my life. Then, it came to me – what’s the one thing I have come to appreciate in the past year? That I’m married to someone who would wake me up by bringing me coffee in bed and talking about the complete works of Euclid!
In my misspent youth, I was the first American to win the world judo championships. This came about since I had a propensity to run my mouth off, which often led to fights. Those people who said I better be able to “walk the walk if I was going to talk the talk”. Well, I took them seriously. In retrospect, they probably wish they had instead advised me to just shut the hell up. Too late now.
In a presentation to our judo board of directors, a marketing specialist commented that a major problem was that materials were developed “with black belt eyes”. In other words, we had pictures of flashy throws by top athletes which made us, the black belts, think the person was extremely skilled and reminded us of when we were young athletes. The market analyst said to us,
The reaction of the average person seeing these is either, “ouch! That looks like it hurts!” or “An older person (or younger person or overweight person or out-of-shape person) like me could never do that.”
For computer science, I think we have the exact same problem. For example, being a decent programmer requires, as a minimum, some basic level of algebra and statistics. You need to understand scientific notation, subscripts, superscripts, a few symbols like ∑ . I’m not talking even Calculus here, but stuff like what the mean of X is and how computation is affected by the distribution of parentheses. This is stuff that a lot of us learned in seventh through tenth grade. What if you didn’t? What if you went to a school where you were taught by seven substitutes in nine months and you never got to it? What if your school didn’t have textbooks? I’m talking about American schools. Probably not the ones your kids go to, if you are reading this, but American schools nonetheless.
Sometimes you have a very good, knowledgeable hard-working teacher and you still didn’t learn it. My older brother is a math teacher and you couldn’t ask for a better one. He often mentions how unmotivated his students are. Let’s think about those black belt eyes for a moment. Most of Algebra is taught completely separate from anything remotely important to the child’s life. I have a daughter in seventh grade at a good school and her textbook has the “appropriate” distribution of different names that is supposed to make it relevant for kids,
Suppose Keisha is making $6 per hour more than Diego. Diego was paid $18 for three hours of working after school at his family’s market. How much money does Keisha make per hour?
We tell kids that they will use algebra in their life and they are skeptical because they don’t really see their parents or clerks at the grocery store working problems out on pieces of paper. When we get to students in community college, they are already calling bullshit on us. That twenty-something student in your Developmental Mathematics class knows that he does not use Algebra in his job at 7-11 and he is pretty sure that his fascist boss doesn’t either.
Don’t even get me started on statistics. Oops, too late!
Sometimes I have to wonder if anyone who writes those statistics textbooks ever met an actual student. Yes, I find puzzling out pages of equations challenging and interesting. Many equations I can glance at and say to myself, “Sum of squares error” and move on. Most people are not me. It goes back to that issue of “black belt eyes” again. Much of what is in an average statistics book – Poisson distributions, the central limit theorem, matrix algebra approaches to multiple regression – students won’t need to know for years, if ever. Several bad things have happened over the years, and I don’t even know where to start.
We have tried to cram so much into textbooks, and into introductory courses, I guess to please whoever thinks we water down the curriculum, that it is virtually impossible to teach it all well in the time allowed.
It is possible, and even rewarded, for students to get by on memorizing formula, facts and sample problems without really understanding what is going on.
Students who come in without the prerequisites are just plain screwed unless the teacher or a TA has the time and kindness to spend hours getting the student up to speed. With cuts in budget and the epidemic of part-time faculty at the college level, let’s just abbreviate it to students who come in without the prerequisites are just plain screwed.
How about this for a completely different idea…. Let’s teach a year of Integrated Computer Science for ten credits each semester. Let’s include programming techniques, algebra and statistics. Let’s have real word problems like deciding how many representatives each state gets and, unlike this Census video, let’s NOT skip over the math. And then let’s write a program to do it. And see if we get the same answers as the U.S. census.
Quit assuming that the only things students are interested in programming are computer games. Quit assuming that students will spend two years struggling in Developmental Mathematics, College Algebra, Statistics and Computer Science 101 courses so that MAYBE in a year or two they will get to write a database program. Read Shelia Tobias’ book “They’re not dumb, they’re different: Stalking the second tier “.
And if you have the attitude that anyone who isn’t willing to learn several semesters’ worth of apparently (to them) useless material, didn’t come into college with all the prerequisites already learned or doesn’t immediately grasp new equations, proofs and concepts has no place in computer science – do us all a favor and don’t go into teaching.